PHYSICAL REVIEW X 7, 021049 (2017) Theory for Transitions Between Exponential and Stationary Phases: Universal Laws for Lag Time Yusuke Himeoka and Kunihiko Kaneko Department of Basic Science, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8902, Japan (Received 11 November 2016; revised manuscript received 25 February 2017; published 27 June 2017) The quantitative characterization of bacterial growth has attracted substantial attention since Monod’s pioneering study. Theoretical and experimental works have uncovered several laws for describing the exponential growth phase, in which the number of cells grows exponentially. However, microorganism growth also exhibits lag, stationary, and death phases under starvation conditions, in which cell growth is highly suppressed, for which quantitative laws or theories are markedly underdeveloped. In fact, the models commonly adopted for the exponential phase that consist of autocatalytic chemical components, including ribosomes, can only show exponential growth or decay in a population; thus, phases that halt growth are not realized. Here, we propose a simple, coarse-grained cell model that includes an extra class of macromolecular components in addition to the autocatalytic active components that facilitate cellular growth. These extra components form a complex with the active components to inhibit the catalytic process. Depending on the nutrient condition, the model exhibits typical transitions among the lag, exponential, stationary, and death phases. Furthermore, the lag time needed for growth recovery after starvation follows the square root of the starvation time and is inversely related to the maximal growth rate. This is in agreement with experimental observations, in which the length of time of cell starvation is memorized in the slow accumulation of molecules. Moreover, the lag time distributed among cells is skewed with a long time tail. If the starvation time is longer, an exponential tail appears, which is also consistent with experimental data. Our theory further predicts a strong dependence of lag time on the speed of substrate depletion, which can be tested experimentally. The present model and theoretical analysis provide universal growth laws beyond the exponential phase, offering insight into how cells halt growth without entering the death phase. DOI: 10.1103/PhysRevX.7.021049 Subject Areas: Biological Physics, Nonlinear Dynamics I. INTRODUCTION others [5–8], in which the constraint to maintain steady growth leads to general relationships [9–11]. Quantitative characterization of a cellular state, in terms of In spite of the importance of the discovery of these the cellular growth rate, concentration of external resources, universal laws, cells under poor conditions exhibit differ- and abundances of specific components, has long been one ent growth phases in which such relationships are violated. of the major topics in cell biology, ever since the pioneering Indeed, in addition to the death phase, cells undergo a study by Monod . Such studies have been developed stationary phase under conditions of resource limitation, in mainly by focusing on the microbial, exponentially growing which growth is drastically suppressed. Once cells enter phase, in which the number of cells grows exponentially the stationary phase, a certain time span is generally (this phase is often termed the log phase in cell biology, required to recover growth after resources are supplied, but considering the focus on exponential growth, here adopt which is known as the lag time. There have been extensive the term “exponential phase” throughout). This work has studies conducted to characterize the stationary phase, uncovered somewhat universal growth laws, including Pirt’s including the length of lag time for resurrection and the equation for yield and growth  and the relationship tolerance time for starvation or antibiotics [12–14],and between the fraction of ribosomal abundance and growth specific possible mechanisms for phase transitions have rate (experimentally demonstrated by Schaechter et al. , been proposed [15–17]. Furthermore, recent experiments and theoretically rationalized by Scott et al. ), among have uncovered the quantitative relationships of lag time and its cell-to-cell variances [18,19]. For example, the lag time was shown to depend on the length of time the cells email@example.com‑tokyo.ac.jp are starved. This implies that the stationary phase is not Published by the American Physical Society under the terms of actually completely stationary but that some slow changes the Creative Commons Attribution 4.0 International license. still progress during the starvation time, in which cells Further distribution of this work must maintain attribution to “memorize” the starvation time. Hence, a theory to explain the author(s) and the published article’s title, journal citation, and DOI. such slow dynamics is needed that can also characterize 2160-3308=17=7(2)=021049(16) 021049-1 Published by the American Physical Society YUSUKE HIMEOKA and KUNIHIKO KANEKO PHYS. REV. X 7, 021049 (2017) is not possible to maintain the population without growth. the phase changes and help to establish corresponding quantitative laws. However, cells often exhibit suppressed growth under The existence of these phases and lag time is ubiquitous substrate-poor conditions, even at a single-cell level in bacteria (as well as most microorganisms). Hence, we [12,13,18], as observed in the stationary phase. Such cells, aim to develop a general model that is as simple as possible, which neither grow exponentially nor move toward death, without resorting to specific detailed mechanisms, but we cannot be modeled with cell models that only consider can nonetheless capture the changes among the lag, autocatalytic processes [4,10,20–24]. exponential, stationary, and death phases. We first describe Therefore, to model a state with such suppressed growth, a simple model for a growing cell, which consists of an it is important to consider additional chemical species, i.e., autocatalytic process driven by active chemical compo- macromolecules that do not contribute to autocatalytic nents such as ribosomes. However, this type of model with growth, in addition to the substrates (S) and component autocatalytic growth from substrates and their derivatives, A(A) that are commonly adopted in models of cell growth. which is adopted for the exponential phase, is not sufficient Component A represents molecules that catalyze their own to represent all phases, as the autocatalytic process either growth, such as ribosomes, and can include metabolic grows exponentially or decays toward death, and thus does enzymes, transporters, and growth-facilitating factors. not account for a halting state with suppressed growth Component B represents waste products or can other corresponding to the stationary phase. Therefore, to go one molecules that are produced with the aid of component step further beyond the simplest model, we then consider A but do not facilitate growth. Thus, the next simplest the addition of an extra class of components that do not model is given by contribute to catalytic growth. Still, even the inclusion of this extra class of components cannot fully account for the dS ¼ −F ðSÞA − F ðSÞA þ AðS − SÞ − μS; A B ext transition to the stationary phase. Therefore, we further dt considered the interaction between the two classes of dA ¼ F ðSÞA − d A − μA; components. Here, we propose a model that includes the A A dt formation of a complex between these two types of dB components, which inhibits the autocatalytic process by ¼ F ðSÞA − d B − μB: ð1Þ B B dt the active components. We show that the model exhibits the transition to the stationary phase with growth suppression. Here, S and S indicate the concentrations of the extrac- ext By analyzing the dynamics of the model, we then uncover ellular and intracellular substrates, respectively. The con- the quantitative characteristics of each of these phases in centration of the intracellular substrate determines the line with experimental observations, including the bacterial synthesis rate of the active and nonautocatalytic proteins growth curve, quantitative relationships of lag time with F and F , respectively. All chemical components are A B starvation time and the maximal growth rate, and the diluted because of the volume growth of a cell. exponentially tailed distribution of lag time. The proposed In addition to dilution, macromolecules (A and B) are model also allows us to derive several experimentally spontaneously degraded with slow rates (d and d ). In this A B testable predictions, including the dependence of lag time model, the cell takes up substrates from the external on the speed of the starvation process. environment from which component A and the non- growth-facilitating component B are synthesized. These A. Model syntheses, S ↔ S, S → A, and S → B, as well as the ext uptake of substrates take place with the aid of catalysis by Since molecules that contribute to autocatalytic proc- component A. Then, by assuming that the synthesized esses are necessary for the replication of cells, models for components are used for growth in a sufficiently rapid growing cells generally consist of at least substrates (S) and period, the growth rate is set to be proportional to the active components (denoted as “component A” hereafter) synthesis rate of component A. Hence, the dilution rate μ of that catalyze their own synthesis as well as that of other each component due to cell volume growth is set as components. For example, in the models developed by μ ¼ F A. Scott et al.  and Maitra et al. , component A Now, if the ratio F =F does not depend on the substrate corresponds to ribosomes, whereas several models involv- A B concentration S, the fraction A=B does not depend on S ing catalytic proteins have also been proposed [10,21–24]. This class of models provides a good description of the either, and the model is reduced to the original autocatalytic exponential growth of a cell under the condition of model; thus, the phase change to suppressed growth is sufficient substrate availability; however, once the degra- not expected. Then, by introducing the S dependence of dation rate of component A exceeds its rate of synthesis F =F to reduce the rate of component A with the decrease A B under a limited substrate supply, the cell’s volume will in the substrate condition, we first tested whether the shrink, leading to cell death. Hence, a cell population either transition to a suppressed growth state, as in the stationary grows exponentially or dies out, and in this cellular state, it phase, occurs under a substrate-poor condition, by setting 021049-2 A THEORY FOR TRANSITIONS BETWEEN EXPONENTIAL … PHYS. REV. X 7, 021049 (2017) (a) F =F to decrease in proportion to the change in S (i.e., A B ½d=ðdSÞF =F > 0). However, in this case, it is straight- A B Component A (A) forwardly confirmed that there is no transition to a sup- F (S) A Growth ( µ ) k A B k C pressed growth state. In other words, the cells always grow p m Substrate( S ) exponentially without any slowing-down process, as the External decrease in S simply influences the growth rate μ, while the Substrate A-B Complex ( C ) F (S) A Component B (B) ( S ) B presence of B does not influence the dynamics of A (see ext also Appendix A). (b) 1 0.1 Thus, we need to introduce an interaction between : µ component A and the non-growth-facilitating component : A B. Although complicated interactions that may involve -5 other components could be considered, the simplest and -3 most basic interaction that can also provide a basis for considering more complex processes would be the for- mation of a complex between A and B given by the reaction -9 -6 Death Inactive Active 10 A þ B ↔ C. This results in the inhibition of the autocat- 0 0 alytic reaction for cell growth, as complex C does not -4 -2 2 4 10 10 1 10 10 contribute to the activity for the autocatalytic process. A ext schematic representation of the present model is shown in (c) (d) Fig. 1(a). Thus, our model is given by Lag Exp. Stationary Stationary Death dS ¼ −F ðSÞA − F ðSÞA þ AðS − SÞ − μS; A B ext dt dA ¼ F ðSÞA − GðA; B; CÞ − d A − μA; A A 2 8 dt (×10 ) (×10 ) -2 0 2 4 0 0.5 1 1.5 2 dB Time ¼ F ðSÞA − GðA; B; CÞ − d B − μB; B B dt FIG. 1. (a) Schematic representation of the components and dC ¼ GðA; B; CÞ − d C − μC; ð2Þ C reactions in the present model. The concentration of each dt chemical changes according to the listed reactions. In addition, chemicals are spontaneously degraded at a low rate, and they where GðA; B; CÞ denotes the reaction of complex for- become diluted because of the volume expansion of the cell. mation, given by k AB − k C. The catalytic activity of p m (b) Steady growth rate and the concentration of component A are component A is inactivated because of the formation of plotted as functions of the external concentration of the substrate. complex C. Here, the complex has higher stability than that (c,d) Growth curve of the model. Parameters are set as follows: of other proteins (d is smaller than d and d ) . −6 −5 C A B v ¼ 0.1, k ¼ 1.0, k ¼ 10 , K ¼ 1.0, K ¼ 10.0, d ¼ d ¼ 10 , p m t R B _ _ From Eq. (2), by summing up A and C, we obtain −12 d ¼ 10 . The detailed numerical method for panels (c) and A þ C ¼ F ðSÞAð1 − ðA þ CÞÞ if d and d are zero (or A A C (d) is given in Appendix C. negligible). This means that once the cell reaches any steady state, the relationship A þ C ¼ 1 is satisfied as long as A and F ðSÞ are not zero. We use this relationship and substrates [32,33]. Therefore, the performance of eliminate C by substituting C ¼ 1 − A for the following these mechanisms is inevitably reduced in a substrate analysis. (energy source)-poor environment. Thus, it naturally One plausible and straightforward interpretation of B is follows that the ratio of the synthesis of active proteins as misfolded or mistranslated proteins that are produced to wastes is an increasing function of the substrate erroneously during the replication of component A. Such concentration, i.e., ½d=ðdSÞf½F ðSÞ=F ðSÞg > 0.In A B waste molecules often aggregate with other molecules the present model, we assume that this ratio increases [26–28]. Alternatively, B components can be specific with the concentration and becomes saturated at higher molecules such as HPF and YfiA [29–31], which inhibit concentrations, as in Michaelis-Menenten’s form, and we catalytic activity by reacting with component A. choose F ðSÞ¼½vS=ðK þ SÞ½S=ðK þ SÞ and F ðSÞ¼ A t B With regards to the formation of error or “waste” ½vS=ðK þ SÞ½K =ðK þ SÞ, for example. t t proteins, there are generally intracellular processes for Note that almost all the results presented in this article reducing their fraction. These include kinetic proofread- are obtained as long as F ≫ F holds for the nutrient-rich A B ing, molecular chaperones, and protease systems. These condition and F ≪ F for the nutrient-poor condition (see A B Sec. II G and Ref. ). Under this condition, a specific error-correction or maintenance systems are energy demanding and require the nonequilibrium flow of choice of the form of F and F is not important. A B 021049-3 Log ( Biomass) µ YUSUKE HIMEOKA and KUNIHIKO KANEKO PHYS. REV. X 7, 021049 (2017) condition F ¼ d . Hence, if d is set to zero, the This S dependence of F =F would be biologically A A A A B inactive-death transition does not occur. plausible for the interpretations of component B as specific We now consider the time series of biomass (the total inhibitory proteins or “waste” (mistranslation) proteins. For amount of macromolecules) that is almost proportional the first interpretation, such proteins related with the to the total cell number, under a condition with a given stationary phase (HPF, YfiA, and others) are induced under finite resource, which allows for direct comparison with stress conditions such as starvation [29,30,35,36]; thus, it is experimental data obtained in a batch culture condition suggested that F ≫ F (F ≪ F ) for a large (small) A B A B [Figs. 1(c) and 1(d)]. To compute the time series of amount of S, respectively. On the other hand, by adopting biomass, we used a model including the dynamics of the latter (waste) interpretation, we derive F ðSÞ and F ðSÞ A B S in addition to S, A, B, and C. Details of this model are close to the above Michaelis-Menten’s form, by consider- ext shown in Appendix C. In the numerical simulation, the ing a proofreading mechanism to reduce the mistranslation condition with a given finite amount of substrates corre- (see also Appendix B). sponding to the increase of cell number is implemented by Here, we also note that, although the S dependence of introducing the dynamics of the external substrate concen- F =F is relevant to derive quantitative laws on the lag A B tration into the original model. Here, S is decreased as the ext time in agreement with experimental observation, it is not substrates are replaced by the biomass, resulting in cell required just to show a transition to a suppressed growth growth. At the beginning of the simulation, the amount of state, as briefly discussed later (see Sec. III). biomass (i.e., cell number) stays almost constant and then gradually starts to increase exponentially. After the phase of II. RESULTS exponential growth, the substrates are consumed, and the A. Growth phases biomass increase stops. Then, over a long time span, the biomass stays at a nearly constant value until it begins to The steady state of the present model exhibits three slowly decrease. Finally, the degradation dominates, and distinct phases as a function of the external substrate the biomass (cell number) falls off dramatically. concentration S [Fig. 1(b)], as computed by its steady- ext These successive transitions in the growth of biomass state solution. The three phases are distinguished by both [Figs. 1(c) and 1(d)] from the initially inactive phase to the the steady growth rate and the concentration of component active, inactive, and death phases correspond to those A, which are termed as the active, inactive, and death observed among the lag, exponential, stationary, and death phases, as shown in Fig. 1, whereas the growth rate shows a phases. As the initial condition was chosen as the inactive steep jump at the boundaries of the phases. The phases are phase under a condition of rich substrate availability, most characterized as follows. (i) In the active phase, the highest of the component A molecules are arrested in a complex at growth rate is achieved, where there is an abundance of this point. Therefore, at the initial stage, dissociation of the component A molecules, which work freely as catalysts. complex into component A and component B progresses, (ii) In the inactive phase, the growth rate is not exactly zero and biomass is barely synthesized, even though a sufficient but is drastically reduced by several orders of magnitude and plentiful amount of substrate is available. After the cell compared with that in the active phase. Here, almost all of escapes this waiting mode, catalytic reactions driven by the component A molecules are arrested through complex component A progress, leading to an exponential increase formation with component B, and their catalytic activity is in biomass. Subsequently, the external substrate is depleted, inhibited. (iii) At the death phase, a cell cannot grow, and and cells experience another transition from the active to all of the components A, B, and complexes go to zero. In the inactive phase. At this point, the biomass only decreases this case, the cell goes beyond the so-called “point of no slowly, owing to the remaining substrate and the stability of return” and can never grow again, regardless of the amount the complex. However, after the substrate is depleted and of increase in S , since the catalysts are absent in any ext components A and B are dissociated from the complex, the form. (As will be shown below, the active and inactive biomass decreases at a much faster rate, ultimately entering phases correspond to the classic exponential and stationary the death phase. phases; however, to emphasize the single-cell growth In the active phase with exponential growth, the present mode, we adopt these former terms for now.) model exhibits classical growth laws, namely, (i) Monod’s The transition from the active to the inactive phase is growth law and (ii) the growth rate vs ribosome fraction caused by the interaction between components A and B. In (see Fig. 6). the substrate-poor condition, the amount of component B exceeds the total amount of catalytic proteins (A þ C), and B. Lag-time dependency on starvation time any remaining free component A vanishes. Below the T and maximum growth rate μ stv max transition point from the inactive phase to the death phase, the spontaneous degradation rate surpasses the synthesis In this section, we uncover the quantitative relationships rate, at which point all of the components decrease. This among the basic quantities characterizing the transition transition point is simply determined by the balance between the active and inactive phases: i.e., lag time, 021049-4 THEORY FOR TRANSITIONS BETWEEN EXPONENTIAL … PHYS. REV. X 7, 021049 (2017) starvation time, and growth rates. We demonstrate that the C. Relationship between lag time pﬃﬃﬃﬃﬃﬃﬃﬃ theoretical predictions agree well with experimentally and starvation time: λ ∝ T stv pﬃﬃﬃﬃﬃﬃﬃﬃ observed relationships. We found that λ increases in proportion to T ,as stv First, we compute the dependence of lag time (λ)on shown in Fig. 2(a). For comparison, the experimentally starvation time ðT Þ. Up to timet ¼ 0, the model cell is set in stv observed relationship between λ and T is also plotted in stv rich a substrate-rich condition, S ¼ S , and it stays at a steady ext ext Fig. 2(b), using reported data [12,18,40] that also exhibited pﬃﬃﬃﬃﬃﬃﬃﬃ state with exponential growth. Then, the external substrate is λ ∝ T dependence. Although this empirical depend- stv poor depleted to S ¼ S instantaneously. The cell is exposed ext ext ence has been discussed previously , its theoretical to this starvation condition up to starvation time t ¼ T . stv origin has not been uncovered thus far. pﬃﬃﬃﬃﬃﬃﬃﬃ Subsequently, the substrate concentration S instantane- ext Indeed, the origin of λ ∝ T can be explained by stv rich ously returns to S . After the substrate level is recovered, it ext noting the anomalous relaxation of the component B takes a certain amount of time for a cell to return to its original concentration, which is caused by the interaction between growth rate (Fig. S1 of Ref. ), which is the lag time λ components A and B. A general description of this following the standard definition of lag time as the time explanation is given below, and the analytic derivation is period before the specific growth rate reaches its maximum given in Ref. . value introduced by Penfold and Pirt [37,38]. Given this, the First, consider the time series of chemical concentrations dependence of λ on the starvation time T can be computed. stv during starvation. In this condition, cell growth is inhibited Next, we compute the dependence of the lag time λ on by two factors: substrate depletion and inhibition of the μ . We choose the steady-state solution of the cell model max catalytic activity of component A. Following the decrease poor under S ¼ S as the initial condition and compute the in uptake due to depletion of S , the concentration of S ext ext ext rich lag time λ under the S ¼ S condition against different decreases, resulting in a change in the balance between A ext ext values of μ ð¼vÞ (following the standard method to and B (hereafter, we adopt the notation such that A, B, and max measure the relationship between λ and μ ). C also denote the concentrations of the corresponding max (a) (b) Slope=0.5 6 0 Slope=0.5 -1 6 9 12 1 2 3 4 Log (T ) Log (Preculture Time) (hours) 10 stv (c) (d) -7 Slope=1.0 Slope=1.0 -8 -1 -2 -9 0 1 2 -1 0 Log ( ) Log (µ ) (1/hours) 10 max 10 max FIG. 2. (a,b) Lag time as a function of (a) starvation time or (b) preincubation time. The lag time is scaled by the maximum growth rate (inversely proportional to the shortest doubling time in the substrate-rich condition). Purple pentagons, cyan dots, and orange squares are adopted from Figs. 3, 6(a), and 6(b) of Augustin et al. , respectively, and the red triangles are extracted from the data in Table 1 of Pin et al. . (c,d) Relationship between the lag time and maximum specific growth rate μ . Data are adopted from Table 1 of Oscar . max poor rich 4 −2 −6 Parameters were set as follows: S ¼ 10 , S ¼ 10 , v ¼ 0.1, k ¼ 1.0, k ¼ 10 , K ¼ 1.0, K ¼ 10.0, and d ¼ d ¼ d ¼ 0 ext ext p m t A B C (the same parameter values as in Fig. 1 except d s). The lag time is computed as the time needed to reach the steady state under the rich S ¼ S condition from an initial condition in the inactive phase. In panel (c), it is obtained by varying vð¼ μ Þ. ext max ext 021049-5 Log (1/ ) Log ( ) 10 10 Log ( µ ) Log (1/ ) (hours) 10 max 10 YUSUKE HIMEOKA and KUNIHIKO KANEKO PHYS. REV. X 7, 021049 (2017) poor chemicals). Under the S condition, the ratio of the lim B∼ lim F · B=B ¼ μ ext A max rich rich S →∞ S →∞ ext ext synthesis of B to A increases. With an increase in B, A decreases because of the formation of a complex with B. holds because lim F ðSÞ¼ μ is satisfied. Thus, it S→∞ A max Over time, more A becomes arrested, and the level of follows that λ ∝ 1=μ . max inactivation increases with the duration of starvation. We also obtained an analytic estimation of the lag time as In this scenario, the increase of the concentration of B is slow. Considering that the complex formation reaction qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A þ B ↔ C rapidly approaches its equilibrium, i.e., λ ∼ 2F k =k T ð3Þ B p m stv max k AB ∼ k C, then A is roughly proportional to the inverse p m of B (recall A þ C ¼ 1)if B is sufficiently large. (see Ref.  for conditions and calculations). In this form, pﬃﬃﬃﬃﬃﬃﬃﬃ Accordingly, the synthesis rate of B, given by F ðSÞA, the two relationships λ ∝ T and λ ∝ 1=μ are stv max is inversely proportional to its amount, i.e., integrated. BðtÞ ∝ F ðSÞ=B; E. Dependence of lag time on the starvation process and thus, So far, we have considered the dependence of lag time on the starvation time. However, in addition to the starvation dB =dt ∼ const: period, the starvation process itself, i.e., the speed required to reduce the external substrate, has an influence on the Hence, the accumulation of component B progresses with pﬃﬃ lag time. BðtÞ ∝ t. (Note that because of S depletion, the dilution For this investigation, instead of the instantaneous effect is negligible). depletion of the external substrate, its concentration is Next, we consider the time series for the resurrection instead gradually decreased over time in a linear manner after recovery of the external substrate. During resurrection, over the span T , in contrast to the previous simulation dec A is increased while B is reduced. Since component A is procedure, which corresponds to T ¼ 0. Then, the cell is dec strongly inhibited after starvation, the dilution effect from placed under the substrate-poor condition for the duration cell growth is the only factor contributing to the reduction T before the substrate is recovered, and the lag time λ is stv of B. Noting that μ ¼ F A and A ∝ 1=B, the dilution effect computed . is given by μB ¼ F AB ∝ B=B ¼ const at the early stage The dependence of the lag time λ on T and T is stv dec of resurrection. Thus, the resurrection time series of B is shown in Fig. 3(a). While λ monotonically increases against determined by the dynamics T for a given T , it shows a drastic dependence on T . stv dec dec If the external concentration of the substrate is reduced BðtÞ ∝−const; quickly (i.e., a small T ), the lag time is rather small. dec However, if the decrease in the external substrate concen- leading to the linear decrease of B, i.e., BðtÞ ∼ tration is slow (i.e., a large T ), the lag time is much dec Bð0Þ − const × t. longer. In addition, this transition from a short to long lag Let us briefly recapitulate the argument presented so far. time is quite steep. The accumulated amount of component B is proportional to pﬃﬃﬃﬃﬃﬃﬃﬃ The transition against the time scale of the environmental T , whereas during resurrection, the dilution of B stv change manifests itself in the time series of chemical progresses linearly with time, which is required for the concentrations [see Fig. 3(b)]. With rapid environmental dissociation of the complex of A and B, leading to growth change, S decreases first, whereas with slow environmental recovery. By combining these two estimates, the lag time pﬃﬃﬃﬃﬃﬃﬃﬃ change, component A decreases first. In addition, the value satisfies λ ∝ T . stv of component B is different between the two cases, indicating that the speed of environmental change affects D. Relationship between the lag time the degree of inhibition, i.e., the extent to which component and maximal growth rate: λ ∝ 1=μ max A is arrested by component B to form a complex. The relationship λ ∝ 1=μ is obtained by numerical Now, we provide an intuitive explanation for two distinct max simulation of our model, in line with experimental results inhibition processes. When S starts to decrease, a cell is ext  [Figs. 2(c) and (d)]. in the active phase in which A is abundant. If the The relationship λ ∝ 1=μ can also be explained by the environment changes sufficiently quickly, there is not max characteristics of the resurrection time series. The dilution enough time to synthesize chemicals A or B because of rate of B over time is given by μB, as mentioned above; the lack of S, and the concentrations of chemical species are thus, at the early stage, B∼−μB. In the substrate-rich frozen near the initial state with abundant A. However, if condition, the substrate abundances are assumed to be the rate of environmental change is slower than that of the saturated, so chemical reaction, the concentration of B (A) increases 021049-6 THEORY FOR TRANSITIONS BETWEEN EXPONENTIAL … PHYS. REV. X 7, 021049 (2017) T = 1 T = 10 (b) (c) dec dec 4 0 (a) 8 6 10 10 2 -2 'result.dat' u 2:1:3 -4 7 5 10 10 -2 -6 0 2 4 6 0 2 4 6 6 4 Log (t) 10 10 Log (t) (d) (e) 5 3 0 10 10 -2 4 2 2 10 10 3 4 5 6 7 10 10 10 10 10 -4 dec 0 0 2 4 6 0 2 4 Log (t-T ) Log (t) 10 stv FIG. 3. (a) Dependence of lag time λ on the time required to decrease the substrate T and starvation time T .(b–d) Time series of dec stv starvation for different T [T ¼ 10 (green line) and T ¼ 1.0 (purple line)] values—the internal concentrations of substrate S (b), dec dec dec component A (c), and component B (d). (e) Time series of biomass during resurrection. The same parameter values as indicated in Fig. 2 were adopted. The batch culture model (which is used to compute a bacterial growth curve) was adopted to compute the time series of biomass accumulation (e). The time series of μ is shown in Fig. S3 of Ref. . (decreases). Hence, A remains rich in the case of fast In contrast, for a slow change (i.e., large T ), the flow dec environmental change, whereas B is rich for a slow in ðA; BÞ gradually changes as shown in Figs. 4(b)–4(d). environmental change. In the former case, when the Initially, the state ðA; BÞ stays at the substrate-rich steady substrate is increased again, component A molecules are state. Because of the change in substrate concentration, ready to work, so the lag time is short, which can be two nullclines moderately move and interchange their interpreted as a kind of “freeze-dry” process. Note that the vertical locations. Since the movement of nullclines is difference in chemical concentration caused by different slow, the decrease in A progresses before the two null- T values is maintained for a long time because, in the clines come close together (i.e., before the process is dec case of slow (fast) environmental change, chemical reac- slowed down). The temporal evolution of A and B is tions are almost completely halted because of the decrease slowed down only after this decrease in A [Figs. 4(c) and of A (S). Thus, the difference of lag time remains even for 4(d)]. Hence, the difference between cases with small and large T , as shown in Fig. 3(a). large T is determined according to whether the null- stv dec This lag time difference can also be explained from the clines almost coalesce before or after the A decrease, perspective of dynamical systems . For a given S, respectively. the temporal evolution of A and B is given by the flow in These analyses allow us to estimate the critical time the state space of ðA; BÞ. Examples of the flow are given in for a substrate decrease T beyond the point at which λ dec Fig. 4. The flow depicts ðdA=dt; dB=dtÞ, which determines increases dramatically. The value of a fixed point ðA ;B Þ st st the temporal evolution. The flow is characterized by A- and depends on the substrate concentration, which drastically B-nullclines, which are given by the curves satisfying changes at the active-inactive transition point. If the dA=dt ¼ 0 and dB=dt ¼ 0, as plotted in Fig. 4. relaxation to the fixed point is faster than the substrate Note that at a nullcline, the temporal change of one state decrease T , the system changes “adiabatically” to follow dec variable (either A or B) vanishes. Thus, if two nullclines the fixed point at each substrate time during the course of a approach each other, then the time evolution of both concen- “slow decrease.” The relaxation time is estimated by the trationsA andB slow down, and the point where two nullclines smallest eigenvalue around the fixed point at the transition intersect corresponds to the steady state. As shown in Fig. 4, point. In the k → 0 limit, this eigenvalue is equal to the nullclines come close together under the substrate-depleting growth rate at the active-inactive transition point. Since condition, which provides a dynamical system account of the it is inversely proportional to v, the critical time T for dec slow process in the inactive phase discussed so far. the substrate decrease is estimated as T ∝ 1=v. This dec For a fast change [i.e., small T , Fig. 4(a)], S is quickly dec dependence was also confirmed numerically (see Fig. S4 reduced at the point where the two nullclines come close in Ref. ). together. First, B reaches the B nullcline quickly. Then, the state changes along the almost-coalesced nullclines where F. Distribution of lag time the dynamics slow down. Thus, it takes a long time to decrease the A concentration, so at the resumption of the So far, we have considered the average change of chemical concentrations using the rate equation of chemical substrate, a sufficient A can be utilized. 021049-7 stv Log (B) Log (S) Log (Biomass) 10 Log (A) 10 YUSUKE HIMEOKA and KUNIHIKO KANEKO PHYS. REV. X 7, 021049 (2017) -2 5 (a) Fast Decrease, S = 10 (b) t = 9.98×10 , S = 12.0 Slow Decrease 2×10 -2.5 -2.5 A nullcline B nullcline -1 Log (v ) t = 0 10 -5 -5 -6 -4 -2 0 -6 -4 -2 0 5 6 -2 (c) t = 9.99×10 , S = 6.5 (d) t = 2×10 , S = 1.18×10 -2.5 0 -2.5 -1 -5 -5 -3 -3 -6 -4 -2 0 -6 -4 -2 0 Log ( A ) FIG. 4. Movement of nullclines and time evolution of state variables [circles within the state space ðA; BÞ]. (a) The case of a fast substrate decrease (the orange line indicates the orbit, and numbers in white boxes indicate the time points). The orbit of a slow substrate decrease is also plotted (black dashed line). (b–d) The case of a slow substrate decrease. Each point is the value of the state variable at the indicated time and substrate concentration. The vector field v ¼ðdA=dt; dB=dtÞ is also depicted. Parameters are identical to those described in Fig. 2. reactions. However, a biochemical reaction is inherently becomes zero because it is inhibited by component B. stochastic; thus, the lag time is accordingly distributed. When the number of component A molecules becomes This distribution was computed by carrying out a stochastic zero, the only reaction that can take place is a dissociation simulation of chemical kinetics using the Gillespie algo- reaction (C → A þ B). Since we assume that the time rithm . evolution of molecule numbers follows a Poisson process, By increasing the starvation time, two types of lag-time the queueing time of dissociation obeys an exponential distributions are obtained: (1) a skewed type and (2) a distribution Probðqueueingtime¼tÞ∼N k expð−N k tÞ, C m C m skewed type with an exponential long-time-tail type. Each where N is the number of complexes formed. This distribution type changes as follows: First, when the exponential distribution is added to the skewed distribution, starvation time is sufficiently long, the system enters the resulting in a long tail. phase with the slow accumulation of B. Here, the relaxation The distributions of the two cases are plotted in Fig. 5, is anomalous, leading to a skewed-type distribution. This together with experimental data adopted from Ref. . skewed distribution is understood as follows. The number The skewed distribution fits the experimental observations of component A molecules among cells takes on a for the 0-day starvation data, whereas the distribution Gaussian-like distribution just before the recovery of the including the exponential tail is a good fit to the 1-day, external substrate concentration , whereas the lag time 2-day, and 3-day distributions. λ is proportional to B and thus to 1=A, as discussed in the Here, each kinetic parameter alters the critical starvation last section. Then, the lag-time distribution λ is obtained as time around which the shape of the distribution starts to the transformation of 1=A → λ from the Gaussian distri- change; for example, a small k makes it easier to obtain bution of component A. This results in a skewed distribu- the type-three distribution. However, kinetic parameters do tion with a long time tail as shown in Fig. 5(a). Second, not change the shape of the distribution directly, as when the starvation time is too long, the decrease in A confirmed computationally. comes to the stage where its molecular number reaches 0 or The distribution of lag time was traditionally thought to 1. This results in a long time tail in the distribution. This follow the normal distribution [8,47] until single-cell measurements were carried out for a long time span effect occurs when the number of component A molecules 021049-8 Log ( B ) 10 THEORY FOR TRANSITIONS BETWEEN EXPONENTIAL … PHYS. REV. X 7, 021049 (2017) (a) procedures. For example, a cell that regains growth in a 0 day colony ends up dominating the colony; thus, the fluctuation of the shortest lag time governs the behavior. However, -2 identification of a small fraction of bacteria with a long lag time is difficult, owing to the limited capacity of cell -4 tracking (as indicated in Ref. ). -5 0 510 (b) G. Remarks on the choice of parameters 1 day to fit the experimental data Although there are several parameters in the model and -2 the results depend on these values, the basic results on the active-inactive transition, suppression of growth, and -4 quantitative relationships with lag time are obtained for 0 10 30 a large parameter region. Conditions of the parameter (c) values to obtain these main results are given in Ref.  2 days and are summarized in Table I. Here, an important parameter is k , which we assumed to be the smallest -2 among all other parameter values. This choice was made to facilitate analytic calculations, and the condition for k can -4 be relaxed. For example, we plotted the growth rate at 10 20 30 the steady state in Fig. S5 of Ref. , indicating that (d) the active-inactive transition occurs as long as k <k 0 m p holds. 3 days Next, we estimate realistic parameter values such as the -2 value of v from the literature. However, several parameter values could not be estimated directly from experimentally reported data because this would require quantitative -4 0 studies at the stationary phase, which are not currently 10 20 30 Normalized Lag Time available. Thus, we estimated other parameter values by fitting Monod’s growth law , as well as by using the FIG. 5. Distribution of lag time obtained by model simulation reported relationship between the ribosome fraction and (solid line) with experimental data (lag-time distribution of growth rate [4,48,49] (Fig. 6) . Since the number of cultures starved for the indicated days) overlaid. The horizontal parameters is greater than the minimum number required to axis of each distribution was normalized by using its peak point fit the two laws in Fig. 6, the choice of parameter values is ðPeakÞ and the full width at half maximum (FWHM) as not unique. A possible set of parameter values is listed in λ → ðλ − PeakÞ=FWHM. Experimental data were extracted from Table II in Appendix D. those presented in Fig. 1(e) of Reismann et al. . Methods of In fitting the two growth laws in Fig. 6, we have also stochastic simulations, the procedure used to compute Peak and found that v is proportional to the maximum growth rate FWHM, and parameter values are given in Appendix C. and that it negatively correlates with the slope of the linear relationship between ribosome fraction and growth rate, . The preset model also generates the normal distribu- while r (the fraction of actively translating ribosome) tion of lag time if the starvation time is too short, whereas decreases and k increases the y offset of the linear relation, the normal distribution of lag time in earlier experiments respectively . would originate from the limitation of experimental TABLE I. Predictions and assumptions. Result Assumption Condition (Prediction) qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Active-inactive transition point k ∼ 0 act−inact act−inact 1 þ 2F (S ðS Þ)=k ¼ 1 þ 4F (S ðS Þ)=k . A st ext p B st ext p inact−death Inactive-death transition point − F (S ðS Þ) ¼ d A st ext A qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Analytic estimation of lag time k ∼ 0, dynamics poor rich λ ∼ f1=½F ðS ðS ÞÞg 2F ðS ðS ÞÞk =k T A st ext B st ext p m stv of S is faster than ðA; BÞ Contiguity of nullclines − A ðBÞ ∼ fGð0;BÞ=½F ðSÞ − G ð0;BÞg A-nullcline A (slow relaxation) A ðBÞ ∼ fGð0;BÞ=½F ðSÞ − F ðSÞB − G ð0;BÞg B-nullcline B A 021049-9 Frequency φ YUSUKE HIMEOKA and KUNIHIKO KANEKO PHYS. REV. X 7, 021049 (2017) (a) (b) 1 0.16 0.12 0.5 0.08 0.04 0 0 -1 -3 -2 0 0.4 0.8 1.2 10 10 10 1 µ (1/h) S (mM) ext FIG. 6. Comparison of the model results using estimated values (Table II) with experimental values. (a) The specific growth rate is plotted as a function of the external substrate (glucose) concentration. Experimental data are adopted from Monod . (b) Ratio of ribosomal proteins (component A) to total proteins as a function of the specific growth rate: Orange triangles are from Scott et al. , red circles are from Bremer and Dennis , and green squares are from Forchhammer et al. . In panel (b), the theoretical curve from the model is plotted up to about 1.0 because we obtained the parameter values by fitting the μ − ϕ relation and the Monod equation with the maximum growth rate of μ ∼ 1.0. max III. DISCUSSION interpret such nonautocatalytic proteins as specific inhibi- tory molecules binding ribosomes such as YfiA and HPF Here, we developed a coarse-grained model consisting of [29–31]. a substrate, autocatalytic active protein (component A), a For a simpler model, one could eliminate the substrate non-growth-facilitating component (component B), and an dependence of F ðSÞ=F ðSÞ. Indeed, even in this simpler B A A-B complex, C. In the steady state, the model shows distinct form, the active-inactive transition itself is observed if we phases, i.e., the active, inactive, and death phases. In finely tune the parameter values, as the decrease in substrate addition, the temporal evolution of the total biomass is flow decreases the dilution, which in turn increases the consistent with the bacterial growth curve. The present fraction of complexes formed. Nevertheless, the accumu- model not only satisfies the already-known growth laws lation of nonautocatalytic proteins is not facilitated with a in the active phase but also demonstrates two relationships, pﬃﬃﬃﬃﬃﬃﬃﬃ λ ∝ T and λ ∝ 1=μ , concerning the duration of the lag substrate decrease, and the increase in the lag time as stv max pﬃﬃﬃﬃﬃﬃﬃﬃ time λ. Although these two relationships have also been λ ∝ T does not follow. Hence, this simpler model will stv observed experimentally, their origins and underlying mech- not be appropriate to explain the behavior of the present anisms had not yet been elucidated. The present model can cells, although it might provide relevant insight as a general explain these relationships based on the formation of a mechanism for the “inactive” or “dormancy” phase in the complex between components A and B, whose increase in context of protocells. the starvation condition hinders the catalytic reaction. Although the cell state with exponential growth has been The above two laws are also generally derived for the extensively analyzed in previous theoretical models, the inactive phase, which corresponds to the stationary phase, transition to the phase with suppressed growth has thus far as long as the ratio of the synthesis of component B to that not been theoretically explained. Our model, albeit simple, of component A is increased along with a decrease in the provides an essential and general mechanism for this external substrate concentration. This condition can also be transition with consideration of the complex formation interpreted as a natural consequence of the waste-reducing between components A and B, which can be experimen- (or error-correcting) process that is ubiquitous in a cell, which demands energy when assuming that component B tally tested. consists of waste molecules. These laws are also derived if The model here may also be relevant to study growth the waste is interpreted as a product of erroneous protein arrest such as stringent response [54,55]. In this case, synthesis, where a proofreading mechanism to correct the ppGpp, the effector molecule of the stringent response, is error, which also requires energy, works inefficiently in a known to destabilize the open complex of all promoters, substrate-poor condition. The inhibition of growth by waste causing the global reduction of macromolecular synthesis, proteins is experimentally discussed by Nucifora and others playing a similar role as component B in the present paper [26–28]. Aggregation of such waste proteins can inhibit the [56–59]. Additionally, rpoS, the sigma factor of stationary- catalytic activity of proteins, although its role in the phase genes, lies downstream of ppGpp , and it is reported that the mutant lacking ppGpp (which might transition to the inactive phase remains to be elucidated. Alternatively, instead of waste proteins, we can also correspond to inhibition of the component B in our model) 021049-10 µ (1/h) THEORY FOR TRANSITIONS BETWEEN EXPONENTIAL … PHYS. REV. X 7, 021049 (2017) shows a physiological state reminiscent of exponentially AðtÞ¼ ; −1 growing bacteria even under starvation . 1 − expð−F tÞð1 − Að0Þ Þ Moreover, the model predicts that the lag time differs Bð0ÞF 1 − expð−F tÞð1 − Þ F A Að0ÞF depending on the rate of external depletion of the substrate, B B BðtÞ¼ ; −1 which can also be examined experimentally. Recently, the F 1 − expð−F tÞð1 − Að0Þ Þ A A bimodal distribution of growth resumption time from the where we neglect d , as in the main text. Therefore, if the stationary phase was reported in a batch culture experiment model cell, Eq. (A2), restarts growth in a high-S (S ) . The heterogeneous depletion of a substrate due to the rich value environment after exposure to the starvation con- spatial structure of a bacterial colony is thought to be a dition (low-S value), AðtÞ and BðtÞ exponentially converge potent cause of this bimodality, and progress toward to the substrate-rich steady state. Hence, the time for gaining a deeper understanding of this concept is underway. growth recovery T is quite short, which is calculated as Since the present model shows different lag times for rec different rates of environmental change, it can provide a 1 B possible scenario to help explain this bimodality. p −F ðS ÞT A poor stv T ¼ ln − 1 ð1− e Þ þ const; rec F ðS Þ B A rich r ACKNOWLEDGMENTS as a function of starvation time T . Here, B and B are stv p r The authors would like to thank S. Krishna, S. Semsey, the steady concentrations of component B under the N. Mitarai, A. Kamimura, N. Saito, and T. S. Hatakeyama substrate-poor and substrate-rich environments, respec- for useful discussions, and I. L. Reisman, N. Balaban, and tively. Obviously, this relationship is far from the relation- J. C. Augustin for providing data. This research is partially ship between lag and starvation time. supported by the Platform for Dynamic Approaches to Living System from the Japan Agency for Medical APPENDIX B: REDUCTION OF THE KINETIC Research and Development (AMED), Grant-in-Aid for PROOFREADING MODEL Scientific Research (S) [No. 15H05746 from the Japan Society for the Promotion of Science (JSPS)], and JSPS In the main text, the concrete forms of F and F A B Grant No. 16J10031. were predetermined by assuming the characteristic ½d=ðdSÞðF =F Þ > 0, which is essential for the active- A B inactive transition. In this section, we show that this APPENDIX A: MODEL WITHOUT characteristic is derived from a simple polymer elongation INTERACTION BETWEEN THE model with a kinetic proofreading scheme  by TWO COMPONENTS assigning a correct polymer as A and an erroneous one To clarify the necessity of the interaction between the as B. Indeed, ½d=ðdSÞðF =F Þ > 0 originates from an A B two components to obtain the main results, we remove the error in the synthesis of component A that consequently complex formation between A and B (by setting k and k inhibits the synthetic reactions. p m to zero). Then, the A-B complex is eliminated, and our Polymer elongation is essential to synthesize macro- model is given as molecules. It is well known that ribosomes elongate a polypeptide chain following receipt of the information from A ¼ F ðSÞA − F ðSÞA − d A; ðA1Þ messenger RNA. However, since the transfer RNA (tRNA) A A A discrimination by a ribosome is not perfect, there is B ¼ F ðSÞA − F ðSÞAB − d B: ðA2Þ always a certain probability for mistranslation (i.e., the B A B wrong choice of tRNA). Kinetic proofreading is one (We assume that the internal concentration of the substrate of the possible error-correction mechanisms in such a is equal to that of the external concentration of the polymerization system, which demands energy. We find substrate, and ignore the substrate dynamics.) The steady that the synthesis ratio of mistranslated proteins to a solution is “correct” protein increases under the substrate-depleting condition. d 1 − d =F ðSÞ For the polymerization reaction, we introduce two A A A A ¼ 1 − ;B ¼ F ðSÞ ; st st B monomers, “correct” and “wrong” monomers, as simplified F ðSÞ F ðSÞ − d þ d A A A B from real amino acids. In reality, there are 20 amino acids and the steady growth rate is given as μ ¼ F ðSÞA ¼ and one tRNA that specifies one amino acid, i.e., one st A st F ðSÞ − d . Therefore, the present model without an correct and 19 wrong monomers with a certain affinity A A interaction between components A and B exhibits lower than that of the correct monomer. only the active-death transition at S , satisfying In the model, a polymer is elongated up to the length L F ðS Þ¼ d . In addition, the dynamics of the system with the aid of the catalytic activity of the “correct” protein, A A are calculated as i.e., the ribosome. The matured polymer with length L is 021049-11 YUSUKE HIMEOKA and KUNIHIKO KANEKO PHYS. REV. X 7, 021049 (2017) spontaneously folded into a protein; the proteins consisting As in the original model, mistranslated proteins inhibit the of only correct monomers are correct proteins with catalytic correct protein’s catalytic activity by forming a complex activity, whereas those with other monomer sequences turn with it, while the growth is facilitated by the activity of into mistranslated proteins. The elongation process pro- correct proteins. The dynamics of the polymer elongation gresses under a kinetic proofreading mechanism (Fig. 7). part are given by X X d½A ðxÞ ˆ ˆ ˆ ˆ ¼ −k ð½A ðxÞ½M − ρ l ½A ðxÞM Þ − k ð½A ðxÞ½M − ρ l ½A ðxÞM Þ þ v ˆ½A ðx ÞM ; 0 i Y Y 0 i Y 2 i Y Y 2 i i−1 terðxÞ dt Y¼C;D Y¼C;D ð1 ≤ i ≤ L − 1Þ; d½A ðxÞM i Y ˆ ˆ ˆ ˆ ¼ k ð½A ðxÞ½M − ρ l ½A ðxÞM Þ − k ð½A ðxÞM α − l ½A ðxÞM βÞ; ð0 ≤ i ≤ L − 1Þ; 0 i Y Y 0 i Y 1 i Y 1 i dt d½A ðxÞM i Y ˆ ˆ ˆ ˆ ¼ k ð½A ðxÞM α − l ½A ðxÞM βÞþ k ð½A ðxÞ½M − ρ l ½A ðxÞM Þ − v ˆ½A ðxÞMð0 ≤ i ≤ L − 1Þ; 1 i Y 1 i Y 2 i Y Y 2 i Y i Y dt where ½M and ½M denote the concentrations of GDP, respectively, and ρ reflects the difference in affinity C D i correct and wrong monomers, respectively. Note that between the wrong monomer (D) and the correct monomer ½A ðxÞ; ½A ðxÞM , and ½A ðxÞM represent the concen- (C) (we set ρ as unity). i i Y i Y C At the steady state, the synthesis rates of correct and tration of a complex of correct proteins and a polymer with L L length i, a correct protein-polymer-monomer complex, and mistranslated proteins, J and J , are given by an activated correct protein-polymer-monomer complex, L L−1 respectively, where x denotes a monomer sequence such as J ¼ v ˆ½A ½M H Ξ ; 0 C C A C CCDC…, with C and D indicating the correct and wrong L L−1 L J ¼ v ˆ½A ðH ½M þ H ½M ÞðΞ þ Ξ Þ − J ; − 0 C C D D C D B A monomers, respectively. Here, terðxÞ and x indicate the last monomer (C or D) of a monomer sequence x and the where functions Ξ and H are given by i i partial monomer sequence of x from which the last monomer [i.e., terðxÞ] has been removed, respectively. vH ˆ ½M i i Note that ½A denotes the concentration of the correct Ξ ¼ ; P P 0 i ˆ ˆ ˆ ˆ k ð1 − ρ l Z ½M Þ þ k ð1 − ρ l H ½M Þ ˆ 0 j j 0 j j 2 j j 2 j j protein; v ˆ and k ’s are the rate constants of the chemical ˆ ˆ ˆ ˆ ˆ ˆ reactions; and the l ’s are the Boltzman factors of each k ðk α þ k ρ l Þþ k k α 2 1 0 i 0 0 1 H ¼ ; chemical reaction. We assume that dissociation of the 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ðk α þ k ρ l Þðk l β þ k ρ l þ v ˆÞ − k l αβ 1 0 i 0 1 1 2 i 2 1 1 matured polymer from correct proteins and polymer folding ˆ ˆ ˆ k þ k l βH 0 1 1 i into proteins takes place instantaneously. Here, α and β are Z ¼ : ˆ ˆ ˆ the concentration energy currencies, for example, GTP and k ρ l þ k α 0 i 0 1 Energy Currency A (x) A (x) A (x)M A (x)M * ^ ^ i+1 i i j i j k k 1 v k l 0 0 ^ k l 1 1 ^ ^ k l k 2 2 Folding Correct Polymer Correct Proteins (Component A) Folding Wrong Polymers Mistranslated protein (Component B) FIG. 7. Schematic representation of a polymer elongation system with kinetic proofreading. The reactions, other than the synthesis part [F ðSÞA and F ðSÞA], are identical to those of the original model (2). A B 021049-12 A THEORY FOR TRANSITIONS BETWEEN EXPONENTIAL … PHYS. REV. X 7, 021049 (2017) L L Here, Ξ and Ξ denote the rate of polymer elongation with In particular, fðdÞ=d½SgJ =J > 0 holds. Indeed, using 0 1 A B the wrong and correct monomers, respectively. Now, we set this model, we obtained the same active, inactive, and death the functional form of α and monomer concentrations ½M phases, as well as the same growth curve and other L L and ½M to obtain the concrete values of J and J .Itis quantitative laws. As an example, Fig. 9 shows the steady D c B natural to assume that α and ½M are increasing functions growth rate as a function of the external substrate concen- of the internal substrate concentration [S]. Here, we adopt tration ½S . Furthermore, for any L, the same behaviors ext L L a Michaelis-Menten’s type form α ¼½S=ðK þ½SÞ, β ¼ are obtained, as J and J satisfy the condition outlined in A B K =ðK þ½SÞ, and ½M ¼½M ¼½M ½S=ðK þ½SÞ. Sec. II of Ref. . It is also confirmed that the ratio of a a C D S max L L L L Although J ð½SÞ and J ð½SÞ do not completely agree J ð½SÞ to J ð½SÞ increases as [S] increases for any L. A B A B with the form we adopted for F ðSÞA and F ðSÞA in the A B original model, the conditions discussed in Sec. II of Ref.  are nevertheless satisfied, as shown in Fig. 8. APPENDIX C: DETAILS OF MODELS AND SIMULATION PROCEDURES To obtain the growth curve shown in Figs. 1(c) and 1(d) L=10 in the main text, we added the dynamics of the substrates in L=10 the external environment, as well as cell volume growth. By -2 10 representing the dynamics according to the amounts of chemicals rather than their concentrations, the model is given by -4 dN ext ¼ −N ðN =V − N =VÞ; ðC1Þ A S bath S ext dt -6 -4 4 dN 10 1 10 ¼ −F ðN =VÞN − F ðN =VÞN A S A B S P -4 -2 2 4 10 10 1 10 10 dt [S] þ N ðN =V − N =VÞ; ðC2Þ A S bath S ext L¼10 L¼10 FIG. 8. J and J are plotted against the substrate A B L¼10 L¼10 dN concentration [S]. The ratio of J to J is also plotted A B ¼ F ðN =VÞN − k N N =V A S A p A B in the inset of the figure. Parameters for the polymer elongation dt 5 2 ˆ ˆ part are set to v ˆ ¼ 0.1, ρ ¼ 1.0, ρ ¼ 10.0, k ¼ 10 , k ¼ 10 , C D 0 1 þ k N − d N ; ðC3Þ m C A A ˆ ˆ ˆ ˆ k ¼10.0, l ¼l ¼expð−1Þ, l ¼ expð1Þ, K ¼ 10.0, K ¼ 1.0, 2 0 1 2 a S ½M ¼ 1.0, and ½A ¼ 1.0. max 0 dN ¼ F ðN =VÞN − k N N =V B S A p A B dt 1 þ k N − d N ; ðC4Þ m C B B : A dN ¼ k N N =V − k N − d N ; ðC5Þ p A B m C C C dt -3 -5 10 dV ¼ F ðN =VÞN ; ðC6Þ A S A dt -6 where N is the amount of substrate in the external ext -10 environment at volume V , and N , N , N , and N are bath S A B C the amounts of each chemical within the cell at volume ~ VðtÞ, respectively. Here, VðtÞ is the volume of a cell. The ~ ~ ~ ~ ~ ~ 0 dilution effect is introduced by dividing the amount of each -4 -2 2 4 10 10 1 10 chemical by VðtÞ, and S is the total amount of the ext [S] external substrate contained in the culture system with volume V (set to unity). For all other parameters, the bath FIG. 9. Steady growth rate of the model with polymerization L¼10 L¼10 same values as shown in Fig. 1 were adopted. and kinetic proofreading. Here, J and J are adopted for A B To obtain the lag-time distribution, we performed a the synthetic reaction rate of components A and B. Parameters for L¼10 L¼10 stochastic simulation. We computed the model equation J and J are identical to those in Fig. 8, and others are set A B to be the same. according to the volume change 021049-13 Flux L=10 L=10 J A J B YUSUKE HIMEOKA and KUNIHIKO KANEKO PHYS. REV. X 7, 021049 (2017) dN rich equations for a sufficiently long time under the N ext ¼ −F ðN =VÞN − F ðN =VÞN A S A B S A poor dt condition, N suddenly changes to N and is then S S ext ext þ N ðN =V − N =VÞ; ðC7Þ set at this value over the starvation period T . Then, N A S bath S stv S ext ext rich returns to the original value N . The lag time λ is ext dN A computed as the time needed to double the volume from ¼ F ðN =VÞN − k N N =V þ k N ; ðC8Þ A S A p A B m C V , i.e., the volume at which S recovers. The numerical dt 0 ext results indicate that the absolute value of the correlation dN coefficient between V and λ is small. Here, the difference B 0 ¼ F ðN =VÞN − k N N =V þ k N ; ðC9Þ B S A p A B m C in V in cells does not affect the distribution of the lag time. dt 0 Stochastic simulation was carried out using the Gillespie algorithm. Parameter values were set to V ¼ 2 × 10 , dN div ¼ k N N =V − k N ; ðC10Þ 4 −3 p A B m C poor V ¼ 1.0, N rich ¼ 10 , N ¼ 10 , and the others bath dt S S ext ext were the same as those described in Fig. 2. The length 4 6 6 dV of starvation time T was set to 5 × 10 , 10 , 2 × 10 , and stv ¼ F ðN =VÞN : ðC11Þ A S A 7 10 for Figs. 5(a)–5(d), respectively. dt From the lag-time distribution obtained by numerical Here, we introduced cell division and simulated the simulation, we can compute the peak and FWHM values dynamics of only one daughter cell (to reduce the simu- directly. Since the experimental data do not include a lation time). When the cell volume V reaches the division sufficient amount of samples, we applied a smoothing filter volume V , V halves and chemicals are distributed to two to determine the FWHM, while the peak point was div daughter cells in equal probability. After computing these determined directly. APPENDIX D: ESTIMATED PARAMETER VALUES Here, we list the estimated value of parameters in Table II (for the detail of estimation procedure, see Sec. II. G). TABLE II. Estimated parameter values. 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Physical Review X – American Physical Society (APS)
Published: Apr 1, 2017
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